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The exact stability condition for certain class of fractional-order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit... 

In this work, a novel method is proposed to study the BIBO stability of a fractional delay system. The characteristic equation of a fractional delay system with some transcendental terms has infinitely many roots. Applying D-subdivision method and the Rekasius substitution divide one-dimensional parametric space of the time delay to infinite intervals with finite unstable roots. The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Two illustrative examples are presented to confirm the proposed method results  

The governing differential equation and both classical and non-classical boundary conditions of strain gradient bars are derived using variational approach. A closed-form analytical solution is obtained for static torsion and the characteristic equation, which gives the natural frequencies, is derived and analytically solved for the free torsional vibrations of the strain gradient microbars. A fixed-fixed microbar is considered as a specific case to investigate the torsional size-dependent static and free-vibration behavior of strain gradient microbars. The results of the current model are compared to those of the modified couple stress and classical theories  

This paper presents a method to determine the denominator and numerator polynomials of a closed-loop transfer function in order to obtain desired transient and steady-state responses in terms of the overshoot, speed (time needed for the relative error to become less than a specific amount) and the steady-state error when the desired output is a ramp signal. Meanwhile, a controller is designed with the same characteristic equation for the closed-loop system to eliminate step and ramp disturbances  

An assessment of the relative speeds and payload capacities of airborne and waterborne vehicles accentuates a gap that can be usefully filled by a new vehicle concept, making use of both hydrodynamic and aerodynamic forces. A high speed marine vehicle equipped with aerodynamic surfaces (called an AAMV, ‘aerodynamically alleviated marine vehicle’) is one such concept. There are three major modes of motion in the operation of an AAMV including take-off, cruising and landing. However, during take-off, hydrodynamic and aerodynamic problems of an AAMV interact with each other in a coupled manner, which make the evaluation of this phase much more difficult. In this article, at first aerodynamic... 

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems  

This paper investigates the robust stability analysis of fractional-order interval systems with multiple time delays, including retarded and neutral systems. A bound on the poles of fractional-order interval systems of retarded and neutral type is obtained. Then, the concept of the value set and zero exclusion principle is extended to these systems, and a necessary and sufficient condition is produced for checking the robust stability of them. The value set of the characteristic equation of the systems is obtained analytically and, based on it, an auxiliary function is introduced to check the zero exclusion principle. Finally, two numerical examples are given to illustrate the effectiveness... 

This work deals with the robust D-stability test of linear time-invariant ( LTI ) general fractional order control systems in a closed loop where the system and - or the controller may be of fractional order. The concept of general implies that the characteristic equation of the LTI closed loop control system may be of both commensurate and non-commensurate orders, both the coefficients and the orders of the characteristic equation may be nonlinear functions of uncertain parameters, and the coefficients may be complex numbers. Some new specific areas for the roots of the characteristic equation are found so that they reduce the computational burden of testing the robust D-stability. Based on... 

This paper studies fractional-order systems of retarded type with two independent delays, and determines the stability regions in spaces of delays. In this approach, an auxiliary polynomial is employed to calculate all purely imaginary roots of the characteristic equation of the system on the imaginary axis. Since roots of the characteristic equation are continuous with respect to delays, these purely imaginary roots determine the stability regions in delay space. Also, the necessary and sufficient condition for stability independent of delays is developed for the systems. Furthermore, a simple inequality constraint is established to obtain pure imaginary poles of the scalar systems.... 

The effect of a multiphase fluid, including an arbitrary number of liquid phases, and a functionally graded density fluid on the stability of rotating partially-filled cylindrical shells is investigated. The first-order shear shell theory is used for modeling the structural dynamics of the shell and a 2D model is introduced based on the Navier–Stokes equations, for fluid motion. The multiphase and the functionally graded density fluids are arranged according to the mass density in a steady state condition due to centrifugal forces. Using the boundary conditions between liquid phases and the boundary conditions of the fluid on the cylinder wall, the coupled fluid-structure system model is... 

This paper presents a numerical algorithm for BIBO stability testing of a certain class of the so-called fractional-delay systems. The characteristic function of the systems under consideration is a multi-valued function of the Laplace variable s which is defined on a Riemann surface with finite number of Riemann sheets where the origin is a branch point. The stability analysis of such systems is not straightforward because there is no universally applicable analytical method to find the roots of the characteristic equation on the right half-plane of the first Riemann sheet. The proposed method is based on the Rouche's theorem which provides the number of the zeros of a given function in a... 

This paper presents fractional order multi-input multi-output (MIMO) controllers for the robust performance of airplane longitudinal motion. A novel necessary and sufficient criterion is offered by using the value set concept to analyze the robust performance of fractional order MIMO uncertain systems based on the location of the characteristic equation roots. The criterion is applicable to all linear time-invariant systems of commensurate and incommensurate orders with complex coefficients. The obtained results are applied to an uncertain linear model of a business airplane to improve the robust performance of its longitudinal motion by decentralized MIMO output feedback and MIMO state... 

In this paper the vibration of a spinning cylindrical shell made of functional graded material (FGM) made is investigated. After a brief introduction of FG materials, by employing higher order theory for shell deformation, constitutive relationships are derived. In the next step by utilizing energy method and Hamilton's principle governing deferential equation of spinning cylindrical shell is obtained. By making use of the principle of minimum potential energy, the characteristic equation of natural frequencies is derived. After verification of the results, the effect of changing different parameters such as material grade, L/R, h/R, and spinning velocity on the natural frequency are... 

Vibration analysis of a gantry crane subjected to a moving trolley hoisting a swinging object is studied in this paper. The finite element method is used to model the dynamical system. Newmark's and Runge-Kutta techniques are used for direct integration of the equation of motion of the structure and the moving object respectively. The moving suspended load is modeled with a non-linear equation of motion including two degrees of freedom i.e. hoisting in conjunction with swinging. A series of numerical simulations are carried out to provide a comprehensive parametric study. The study is directed to identify dynamic characteristics of the coupled non-linear system as well as to provide... 

Oscillatory condition in LTI dynamic systems is generally expressed as possessing purely imaginary solutions by their characteristic equations. Dealing with the class of fractional order systems, such a condition is equivalently restated as owing complex roots with specific arguments by a polynomial defined based on the system characteristic equation. The degree of this polynomial can be unboundedly high. Consequently, such statement for the oscillatory condition in fractional order systems, which is based on arguments of the roots of a polynomial with an unbounded degree, cannot be viewed as a closed-form expression. To tackle this challenge, this brief introduces an approach to obtain a... 

In this paper, an optimal down sampler is used for Associate Discrete Circuit (ADC) based modeling and simulation of basic switching converters. Characteristic equation of backward Euler based discrete model of the circuit is used to find the value for down sampling. Using ADC modeling, fixed admittance matrix can be achieved for modeling switching converters and using down sampler, additional switch and diode current oscillations are minimized. Real-time digital simulation of buck converter using ADC with down sampler method is implemented on a Field Programmable Gate Array (FPGA) and the results are those expected without additional numerical oscillations. Based on the place-and-route... 

In many medical applications, it is necessary to precisely control a robot to achieve exact positioning. In such cases, some components of robot which had been considered rigid should be modeled with visco-elastic elements to present more exact model of robot, and hence controllers designed under the assumption of rigidity may not accurately control them, especially when time-delay and disturbances have been appeared in the closed-loop system. In this study, we present a new control approach to force such robots to have rigid and exact motions, while they have visco-elastic components. Time-delay in the feedback path defects transient state response, while disturbances change the steady... 

Overwhelming number of control laws has been studied for control of robot manipulators with rigid links and joints. However controllers designed under this assumption may not accurately control the manipulator link due to visco-elastic properties that appear in the link behavior. In this study, a novel approach for robust control of a single-link manipulator is presented to force the link to have rigid motions, while it has visco-elastic behavior. In this regard, initially robot dynamics is extracted, followed by the design of four appropriate controllers through the loop-shaping approach. The obtained model is first represented in state space, however later converted to transfer function...